Let $T:X\to X$ be a linear operator on a Banach space $X$. We know that the spectrum of $T$ is an upper semicontinuous function of $T$ for the uniform convergence: that is, if $T_n:X\to X$ is a sequence of operators, s.t, $T_n\to T$ uniformly then, given an open set $G$ containing the spectrum of $T$, $\sigma(T)$, there must exist $k$ s.t $n>k$ implies that $\sigma (T_n)$ is contained in $G$.

I would like to know if this result remains valid under weaker conditions, like $T_n$ converges to $T$ in the compact parts of $X$.

I'd like to know if properties like spectral gap in the spectrum remain if we weaken the notion of

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    $\begingroup$ Have a look at the paper of Keller and Liverani. They make an assumption that appears naturally in dynamical systems, and prove a continuity result for the spectrum exterior to the essential spectral radius. $\endgroup$ – Anthony Quas Feb 18 '17 at 1:19
  • $\begingroup$ @AnthonyQuas Could you provide me the name of the paper? $\endgroup$ – Eduardo Feb 18 '17 at 1:36
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    $\begingroup$ Stability of the spectrum for transfer operators $\endgroup$ – Anthony Quas Feb 18 '17 at 3:05

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